Grimsby, who sit 21st in the League Two table, are moving towards making a quick appointment as they look to turn things around this season.
They were keen on handing the job to former Cardiff boss Morison and held talks with him about taking over at Blundell Park.
Morison though has rejected the opportunity after he failed to agree terms with the League Two outfit.
Grimsby will now look elsewhere and it is suggested that an appointment is now imminent at the club.
Morison took over at Cardiff initially in a caretaker capacity before being handed the job on a permanent basis.
He was sacked by the Welsh giants in September 2022 after a poor start to the campaign.
Morison is currently in charge of non-league outfit Hornchurch, but his stock remains high and he could be in line for a return to the EFL sooner rather than later.
In total, Morison took charge of 45 games as Cardiff boss, winning 16 and losing 21.
